# Which complete bipartite graphs are Hamiltonian?

## Which complete bipartite graphs are Hamiltonian?

The complete bipartite graph Kn,n is Hamiltonian, for all n ≥ 2. We note here that for n = 1 or 2, Kn,n is a tree, and is therefore not Hamiltonian.

How many Hamiltonian cycles are there in a complete bipartite graph?

A Hamiltonian graph must contain at least one Hamiltonian cycle.

What makes a complete bipartite graph?

A complete bipartite graph is a graph whose vertices can be partitioned into two subsets V1 and V2 such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph.

### How do you find the complete bipartite graph?

A graph G = (V, E) is called a complete bipartite graph if its vertices V can be partitioned into two subsets V1 and V2 such that each vertex of V1 is connected to each vertex of V2. The number of edges in a complete bipartite graph is m.n as each of the m vertices is connected to each of the n vertices.

Is the complete bipartite K2 3 Hamiltonian?

Proposition 2.1 K2,3 is a non-Hamilton graph with minimum number of graphic elements.

How do you know if a graph is Hamiltonian?

A simple graph with n vertices in which the sum of the degrees of any two non-adjacent vertices is greater than or equal to n has a Hamiltonian cycle.

## How many Hamiltonian circuits are in a complete graph?

A complete graph with 8 vertices would have = 5040 possible Hamiltonian circuits. Half of the circuits are duplicates of other circuits but in reverse order, leaving 2520 unique routes.

Are complete graphs Hamiltonian?

Every complete graph with more than two vertices is a Hamiltonian graph. This follows from the definition of a complete graph: an undirected, simple graph such that every pair of nodes is connected by a unique edge. The graph of every platonic solid is a Hamiltonian graph.

Which complete bipartite graph is a complete graph?

Explanation: In graph theory, Moore graph is defined as a regular graph that has a degree d and diameter k. therefore, every complete bipartite graph is a Moore Graph.

### What is bipartite and complete bipartite graph?

By definition, a bipartite graph cannot have any self-loops. For a simple bipartite graph, when every vertex in A is joined to every vertex in B, and vice versa, the graph is called a complete bipartite graph. If there are m vertices in A and n vertices in B, the graph is named Km,n.

Can a bipartite graph be Hamiltonian?

Let G=(A∣B,E) be a bipartite graph. To be Hamiltonian, a graph G needs to have a Hamilton cycle: that is, one which goes through all the vertices of G. As each edge in G connects a vertex in A with a vertex in B, any cycle alternately passes through a vertex in A then a vertex in B. Hence C can not be a Hamilton cycle.

Does K2 3 have a Hamiltonian cycle?

## Can a bipartite graph have a Hamilton cycle if M = N?

This leads to a contradiction since a cycle cannot have repeating vertices. Hence, a complete Bipartite graph K m, n has a Hamilton cycle if and only if m = n. Is this correct? Show activity on this post. A complete bipartite graph K m, n is Hamiltonian if and only if m = n , for all m, n ≥ 2.

How do you prove that a graph is Hamiltonian?

A complete bipartite graph K m, n is Hamiltonian if and only if m = n , for all m, n ≥ 2. Proof: Suppose that a complete bipartite graph K m, n is Hamiltonian. Then, it must have a Hamiltonian cycle which visits the two partite sets alternately.

Is Hamiltonian circuit NP complete?

As a corollary, HAMILTONIAN CIRCUIT is NP-complete for strongly chordal split graphs. On both classes the complexity of the HAMILTONIAN PATH problem coincides with the complexity of HAMILTONIAN CIRCUIT. Further, we show that HAMILTONIAN CIRCUIT is linear-time solvable for convex bipartite graphs.

### What is the difference between Hamiltonian circuit and Hamiltonian path?

More formally, HAMILTONIAN CIRCUIT and HAMILTONIAN PATH are sets of graphs having a HAMILTONian circuit and HAMILTONian path, respectively. HAMILTONian circuit is one of the most famous topics in graph theory.